Project Summary Even in the absence of changing sensory inputs, many networks in the brain exhibit emer- gent dynamics: that is, they display patterns of neural activity that are shaped by the intrinsic structure of the network, rather than modulated by an external input. Such dynamics are be- lieved to underlie central pattern generators (CPGs) for locomotion, oscillatory activity in cortex and hippocampus, and the complex interplay between sensory-driven responses and ongoing spontaneous activity. The goal of this research is to develop a theory of how emergent dynamics can arise solely from the structure of connectivity between neurons. We will do this in the con- text of a simple but fundamentally nonlinear model: the Combinatorial Threshold-Linear Network (CTLN) model. This model has binary synapses and simple, perceptron-like neurons, ensuring that any emergent dynamics arise purely from the structure of connections, as described by a directed graph. Despite its simplicity, the CTLN model captures the full range of nonlinear dynamic behav- iors observed in neural systems within a single model framework. Crucially, the model is also mathematically tractable, allowing us to prove very general results (theorems) that help guide the applications. We have already obtained theoretical results that enable us to reason about the underlying connectivity graph and obtain valid (and intuitive) predictions about the resulting dynamics. Our speci?c aims will extend these results and develop several applications. Specif- ically, we will further develop the theory of emergent dynamics in the CTLN model, completing our classi?cation of ?xed point attractors and investigating the transition to chaos as a function of connection sparsity. We will use the same framework to study pattern generation in small net- works, in order to understand how sequences and complex rhythms emerge from the structure of connectivity. Finally, we will apply our theoretical results to investigate a variety of dynamic phenomena in hippocampus in a single unifying model.